I'm reading this paper:High frequency trading in a limit order book. IN section 3.1, an HJB equatioin was given without any details. Could anyone show how to arrive this equation step by setp? I have basic knowledge about dynamic programming and hjb equation.
Actually I have seen some similar detailed derivation process in a Chinese website(https://zhuanlan.zhihu.com/p/161632470) but I'm still confused with it in some step. The main step I'm confused with is the expansition of J(x,s,q,t), where x(t) is the wealth in dollar at time t, s(t) is the stock price (s(t)=udt+sigma dBt), q(t) is the inventory at time t, which is a jump process represeted by a Poisson process N(t).
I wonder why there are no second order term with respect to x and q in this expansion? I thought the quadratic variation of Nt is not 0, so there should exist a second order term in this expansion, just like the s's term.